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The quantum-mechanical Coulomb propagator in an L² function representation.

ABSTRACT: The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss-Pollaczek quadrature, the Gauss-Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss-Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L[Formula: see text] basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

SUBMITTER: Gersbacher R

PROVIDER: S-EPMC8460653 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:We performed steered molecular dynamics (SMD) and umbrella sampling simulations of Cl- ion migration through the transmembrane domain of a prototypical E. coli CLC Cl-/H+ antiporter by employing combined quantum-mechanical (QM) and molecular-mechanical (MM) calculations. The SMD simulations revealed interesting conformational changes of the protein. While no large-amplitude motions of the protein were observed during pore opening, the side chain rotation of the protonated external gating residue Glu148 was found to be critical for full access of the channel entrance by Cl-. Moving the anion into the external binding site (Sext) induced small-amplitude shifting of the protein backbone at the N-terminal end of helix F. As Cl- traveled through the pore, rigid-body swinging motions of helix R separated it from helix D. Helix R returned to its original position once Cl- exited the channel. Population analysis based on polarized wavefunction from QM/MM calculations discovered significant (up to 20%) charge loss for Cl- along the ion translocation pathway inside the pore. The delocalized charge was redistributed onto the pore residues, especially the functional groups containing π bonds (e.g., the Tyr445 side chain), while the charges of the H atoms coordinating Cl- changed almost negligibly. Potentials of mean force computed from umbrella sampling at the QM/MM and MM levels both displayed barriers at the same locations near the pore entrance and exit. However, the QM/MM PMF showed higher barriers (~10 kcal/mol) than the MM PMF (~2 kcal/mol). Binding energy calculations indicated that the interactions between Cl- and certain pore residues were overestimated by the semi-empirical PM3 Hamiltonian and underestimated by the CHARMM36 force fields, both of which were employed in the umbrella sampling simulations. In particular, CHARMM36 underestimated binding interactions for the functional groups containing π bonds, missing the stabilizations of the Cl- ion due to electron delocalization. The results suggested that it is important to explore these quantum effects for accurate descriptions of the Cl- transport.

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The quantum-mechanical Coulomb propagator in an L² function representation.

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The quantum-mechanical Coulomb propagator in an L2 function representation.

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OmicsDI is part of the ELIXIR infrastructure

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The quantum-mechanical Coulomb propagator in an L² function representation.